**CRM Monograph Series**

Volume: 35;
2014;
152 pp;
Hardcover

MSC: Primary 11; 14;
**Print ISBN: 978-1-4704-1912-7
Product Code: CRMM/35**

List Price: $79.00

Individual Member Price: $63.20

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# The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type

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*Fritz Hörmann*

A co-publication of the AMS and Centre de Recherches Mathématiques

This book outlines a functorial theory of
integral models of (mixed) Shimura varieties and of their toroidal
compactifications, for odd primes of good reduction. This is the
integral version, developed in the author's thesis, of the theory
invented by Deligne and Pink in the rational case. In addition, the
author develops a theory of arithmetic Chern classes of integral
automorphic vector bundles with singular metrics using the work of Burgos,
Kramer and Kühn.

The main application is calculating arithmetic volumes or
“heights” of Shimura varieties of orthogonal type using
Borcherds' famous modular forms with their striking product
formula—an idea due to Bruinier–Burgos–Kühn and
Kudla. This should be seen as an Arakelov analogue of the classical
calculation of volumes of orthogonal locally symmetric spaces by
Siegel and Weil. In the latter theory, the volumes are related to
special values of (normalized) Siegel Eisenstein series.

In this book, it is proved that the Arakelov analogues are related
to special derivatives of such Eisenstein series. This result gives
substantial evidence in the direction of Kudla's conjectures in
arbitrary dimensions. The validity of the full set of conjectures of
Kudla, in turn, would give a conceptual proof and far-reaching
generalizations of the work of Gross and Zagier on the Birch and
Swinnerton-Dyer conjecture.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Table of Contents

# Table of Contents

## The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type

#### Readership

Research mathematicians and graduate students interested in Shimura varieties, Siegel-Weil theory, Borcherds products, Kudla's conjectures, and Arakelov theory.